Convergence to stable laws and a local limit theorem for stochastic recursions

Mariusz Mirek

Mariusz Mirek

Colloquium Mathematicae, Tome 120 (2010), p. 705-720 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider the random recursion $X ₙ^{x} = M ₙ X_{n - 1}^{x} + Q ₙ + N ₙ (X_{n - 1}^{x})$ , where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and $N ₙ (X_{n - 1}^{x})$ is smaller at infinity, than $M ₙ X_{n - 1}^{x}$ . Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $S ₙ^{x} = \sum_{k = 0}^{n} X_{k}^{x}$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

Publié le : 2010-01-01

Zbl 1188.60012

EUDML-ID : urn:eudml:doc:283412

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     title = {Convergence to stable laws and a local limit theorem for stochastic recursions},
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     year = {2010},
     pages = {705-720},
     zbl = {1188.60012},
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Mariusz Mirek. Convergence to stable laws and a local limit theorem for stochastic recursions. Colloquium Mathematicae, Tome 120 (2010) pp. 705-720. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm118-2-21/